Revisiting extensions of regularly varying functions

TL;DR

This paper explores the relationships among specific classes of functions related to regular variation, providing new characterizations and clarifying their connections to O-regularly varying functions.

Contribution

It introduces a new characterization of the classes ${\\cal M}$, ${\\cal M}_\\infty$, and ${\\cal M}_{-\\infty}$, enhancing understanding of their relationship with O-regularly varying functions.

Findings

Established relationships among classes ${\\cal M}$, ${\\cal M}_\\infty$, and ${\\cal M}_{-\\infty}$

Provided a new characterization of these classes

Connected these classes to O-regularly varying functions

Abstract

Relationships among the classes ${\cal M}$, ${\cal M}_\infty$, and ${\cal M}_{-\infty}$ and the class of \emph{O}-regularly varying functions are shown. These results are based on two characterizations of ${\cal M}$, ${\cal M}_\infty$, and ${\cal M}_{-\infty}$ provided by Cadena and Kratz in [7] and a new one given in this note.